My sister asked me something like five years ago to prove that a particular physical problem has a closed form solution. Are there some theorems to prove the existence of closed form solutions? The problem seems to be difficult as often you can make different kind of equations as differential- or integral- or recursive equations for some problems, but here we need a method to prove that no matter we define the equation, we can or can't find the closed form.

The equation appears in http://biology.anu.edu.au/hosted_sites/kokko/Esdale/index.html and https://physics.stackexchange.com/questions/63674/is-there-a-closed-form-solution-to-the-esdale-river-problem .

  • $\begingroup$ This question seems much too broad. There are existence theorems for certain kinds of differential equations, as well as integral equations. The problem to which you linked might result in a differential equation for which you cannot guarantee a solution. Existence can be a hard thing to prove for many physical problems. $\endgroup$ – Adrian Keister Aug 14 '13 at 19:25

I really doubt there is for a general problem. Galois Theory gives a theorem which determines whether a polynomial has a solution in a closed form:

Theorem : $f(x)$, a polynomial, is soluble by radical $\Longleftrightarrow$ Gal$(f)$ is a soluble group.

Soluble by radical means that it can be expressed using the *,/,+,- and $n^\text{th}$ root which I think is what you mean by closed form. Knowing if Gal$(f)$ is soluble is a very hard work for easy functions and about impossible for the rest!

I know this doesn't exactly answer your question but knowing this, you should understand the reason why I say that I seriously don't think that there is such an algorithm or theorem.

  • $\begingroup$ Well, I was looking something bigger theorem which gives Galois theory and Risch's algorithm as a special case. $\endgroup$ – Jaakko Seppälä Aug 14 '13 at 19:34

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